It denotes a hypothetical situation. Smith is using the word 'suppose' to say that he's not necessarily going to call the other guy that afternoon, but that what he's about to say is a proposal. You could replace the last part with "If I were to give you a ring this afternoon and let you know, would that be okay?" and it would mean the same thing.

It means the same as if he said "What if I give you a call...?" or "How about I give you a call...?"

要结合语境去看，前面说了：but i think so，所以suppose在句中指的是很有可能，相当于maybe但更倾向于去长城。suppose to be在口语中很常用

I believe it has a meaning similar to “要不。。。“ (unless I am mistaken in my Chinese!)